On the inequalities of Zygmund and de Bruijn

For the polar derivative \(D_\alpha P(z) =nP(z)+(\alpha-z)P'(z)\) $D_{\alpha }P(z)=nP(z)+(\alpha -z)P^{\prime }(z)$ of a polynomial \(P(z)\) $P(z)$ of degree n, most of the \(L^p\) $L^p$ inequalities available in the literature are for restricted values of \(\alpha\) $\alpha$ , and in this paper we extend few such fundamental results to all of \(\alpha\) $\alpha$ in the complex plane.